Lecture 15 Conformal Parametrisation of Loxodromes

We provide a parametrisation of a loxodrome by three specially arranged cycles. The parametrisation is covariant under fractional linear transformations of the complex plane and naturally encodes conformal properties of loxodromes. Selected geometric examples illustrate the use of parametrisation. Our work extends the set of objects in Lie sphere geometry—circle, lines and points—to the natural maximal conformally-invariant family, which also includes loxodromes.

15.1 Introduction

It is easy to come across shapes of logarithmic spirals, as in Fig. 15.1(a), one can find them on a sunflower, a snail shell or a remote galaxy. This is not surprising since the fundamental differential equation ẏ=λ y, λ∈ ℂ serves as a first approximation to many natural processes.

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(a)  (b)

Figure 15.1: A logarithnic spiral (a) and its image under a fractional linear transformation—loxodrome (b).

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The main symmetries of complex analysis are built on the fractional linear transformation (FLT):

Thus, images of logarithmic spirals under FLT, called loxodromes, as in Fig. 15.1(b) are not rare. Indeed, they appear in many instances from the stereographic projection of a rhumb line in navigation to a preferred model of a Carleson arc in the theory of singular integral operators [46, 40]. Furthermore, loxodromes are orbits of one-parameter continuous groups of FLT of loxodromic type [24]*§ 4.3 [306]*§ 9.2 [330]*§ 9.2.

This setup motivates a search for effective tools to deal with FLT-invariant properties of loxodromes. They were studied from a differential-geometric point of view in many papers [43, 290, 291, 292, 293, 317], see also [270]*§ 2.7.6. In this work we develop a “global” description that matches the Lie sphere geometry framework, see Rem. 3.

The outline of the paper is as follows. After preliminaries on FLTs and invariant geometry of cycles (Section 15.2) we review the basics of logarithmic spirals and loxodromes (Section 12.2). A new parametrisation of loxodromes is introduced in Section 15.4 and several examples illustrate its use in Section 15.5. Section 15.6 frames our work within a wider approach [209, 211, 212], which extends Lie sphere geometry. A brief list of open questions concludes the paper.

15.2 Preliminaries: Fractional Linear Transformations and Cycles

In this section we provide some necessary background in Lie geometry of circles, fractional-linear transformations and the Fillmore–Springer–Cnops construction (FSCc). Regrettably, the latter remains largely unknown in the context of complex analysis despite its numerous advantages. We will have some further discussion of this in Rem. 3 below.

The right way [306]*§ 9.2 to think about FLT (7) is through the projective complex line Pℂ. This is the family of cosets in ℂ²∖ {(0,0)} with respect to the equivalence relation (

) ∼ (

) for all nonzero α∈ℂ. Conveniently ℂ is identified with a part of Pℂ by assigning the coset of (

) to z∈ℂ. Loosely speaking Pℂ=ℂ∪ {∞}, where ∞ is the coset of (

). The pair [w₁:w₂] with w₂≠ 0 gives homogeneous coordinates for z=w₁/w₂ ∈ℂ. Then, the linear map ℂ²→ ℂ²

factors from ℂ² to Pℂ and coincides with (7) on ℂ⊂ Pℂ.

Generic equations of cycles in real and complex coordinates z=x+i y are:

where (k,l,n,m)∈ℝ⁴ and L=l+i n. This includes lines (if k=0), points as zero-radius circles (if l²+n²−mk=0) and proper circles otherwise. Homogeneity of (1) suggests that (k,l,m,n) can be considered as homogeneous coordinates [k:l:m:n] of a point in three-dimensional projective space Pℝ³.

The homogeneous form of the cycle equation (1) for z=[w₁:w₂] can be written¹ using matrices as follows:

From now on we identify a cycle C given by (1) with its 2× 2 matrix (

) , this is called the Fillmore–Springer–Cnops construction (FSCc) . Again, C shall be treated up to the equivalence relation C ∼ tC for all real t≠ 0. Then, the linear action (2) corresponds to the action on 2× 2 cycle matrices by the intertwining identity:

Explicitly, for M∈GL₂(ℂ) those actions are:

where M is the component-wise complex conjugation of M. Note, that the FLT with matrix M (7) corresponds to a linear transformation C↦ M(C):=MCM⁻¹ of cycle matrices in (6). A quick calculation shows that M(C) indeed has real off-diagonal elements as required for a FSCc matrix.

This paper essentially depends on the following result.

Proof. Indeed, we have:

using the invariance of trace.

Note that the cycle product (7) is not positive definite, it produces a Lorentz-type metric in ℝ⁴. Here are some relevant examples of geometric properties expressed through the cycle product:

In general, a combination of (6) and (8) yields that consideration of FLTs in ℂ can be replaced by linear algebra in the space of cycles ℝ⁴ (or rather Pℝ³) with an indefinite metric, see [113] for the latter.

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Figure 15.2: Orthogonal elliptic (green-dashed) and hyperbolic (red-solid) pencils of cycles. The left drawing shows the standard case and the right the generic one, which is the image of the standard pencils under an FLT.

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A spectacular illustration of this approach that we will need later is orthogonal pencils of cycles. Consider a collection of all cycles passing through two different points in ℂ, it is called an elliptic pencil. A beautiful and non-elementary fact of Euclidean geometry is that cycles orthogonal to every cycle in the elliptic pencil fill the entire plane and are disjoint, such a family is called a hyperbolic pencil. This statement is obvious in the standard arrangement when the elliptic pencil is formed by straight lines—cycles passing the origin and infinity. Then, the hyperbolic pencil consists of concentric circles, see Fig. 15.2. For the sake of completeness, a parabolic pencil (not used in this paper) formed by all circles touching a given line at a given point, [198]*Ex. 6.10 provides further extensions and illustrations. See [330]*§ 11.8 for an example of cycle pencils’ appearance in operator theory.

This picture becomes a bit trivial in the language of cycles. A pencil of cycles (of any type!) is a linear span t C₁+(1−t)C₂ of two arbitrary different cycles C₁ and C₂ from the pencil. Again, this is easier to check for standard pencils. A pencil is elliptic, parabolic or hyperbolic depending on which inequality holds [198]*Ex. 5.28.ii:

Then, the orthogonality of cycles in the plane is exactly their orthogonality as vectors with respect to the indefinite cycle product (7). For cycles in the standard pencils this is immediately seen from the explicit expression of the product ⟨ C,C′ ⟩=LL′+LL′−mk′−km′ in cycle components. Finally, linearization (6) of FLT in the cycle space shows that a pencil (i.e., a linear span) is transformed to a pencil and FLT-invariance (8) of the cycle product guarantees that the orthogonality of two pencils is preserved.

15.3 Fractional Linear Transformations and Loxodromes

In aiming for a covariant description of loxodromes we start from the following definition.

The SLS is the solution of the differential equation z′=λ z with the initial value z(0)=± 1 and has the parametric equation z(t)=± e^{λ t}. Furthermore, we obtain the same orbit for λ₁ and λ₂∈ ℂ if λ₁=aλ₂ for real a≠ 0 through a re-parametrisation of the time t↦ a t. Thus, an SLS is identified by the point [ℜ (λ) : ℑ (λ)] of the real projective line Pℝ. Thereafter the following classification is useful:

Informally: a positive SLS unwinds counterclockwise, a negative SLS clockwise. A degenerate SLS is the unit circle if ℑ(λ)≠ 0 and the punctured real axis ℝ∖ {0} if ℜ(λ)≠ 0. If ℜ(λ)=ℑ(λ)=0 then the SLS is the single point 1.

Obviously, a complex affine transformation is an FLT with the upper triangular matrix (

) . Thus, logarithmic spirals form an affine-invariant (but not FLT-invariant) subset of loxodromes. Thereafter, loxodromes (and their degenerate forms—circles, straight lines and points) extend the notion of cycles from the Lie sphere geometry, cf. Rem. 3.

By the nature of Defn. 7, the parameter λ and the corresponding classification from Defn. 6 remain meaningful for logarithmic spirals and loxodromes. FLTs eliminate distinctions between circles and straight lines, but for degenerate loxodromes (ℜ(λ)· ℑ(λ) = 0) we still can note the difference between the two cases of ℜ(λ)≠ 0 and ℑ(λ)≠ 0: orbits of former are whole circles (or straight lines) while latter orbits are only arcs of circles (or segments of lines).

The immediate consequence of Defn. 7 is

As mentioned above, an SLS is completely characterised by the point [ℜ (λ) : ℑ (λ)] of the real projective line Pℝ extended by the additional point [0 : 0]². In the standard way, [ℜ (λ) : ℑ (λ)] is associated with the real value λ′:=2π ℜ (λ)/ℑ (λ) extended by ∞ for ℑ (λ)=0 and with symbol 0/0 for the ℜ(λ)=ℑ(λ)=0 cases. Geometrically, a=exp(λ′)∈ℝ₊ represents the next point after 1, where the given SLS branch meets the real positive half-axis after one full counterclockwise turn. Obviously, a>1 and a<1 for positive and negative SLS, respectively. For a degenerate SLS:

1. with ℑ(λ)≠ 0 we obtain λ′=0 and a=1;
2. with ℜ(λ)≠ 0 we consistently define a=∞.

In essence, a loxodrome Λ is defined by the pair (λ′, M), where M is the FLT mapping Λ to the SLS with parameter λ′. While λ′ is completely determined by Λ, the map M is not.

15.4 Three-cycle Parametrisation of Loxodromes

Although pairs (λ′, M) provide a parametrisation of loxodromes, the following alternative is more operational. It is inspired by the orthogonal pairs of elliptic and hyperbolic pencils described in Section 15.2.

We say that a three-cycle parametrisation is standard if C₁ is the real axis and C₂ is the unit circle, then C₃={z: | z |=exp(λ′)}. A three-cycle parametrisation can be consistently extended to a degenerate SLS Λ as follows:

• λ′=0: any straight line C₁ passing the origin and the unit circle C₂=C₃=Λ;
• λ′=∞: the real axis as C₁=Λ, the unit circle as C₂ and C₃=(0,0,0,1) being the zero-radius circle at infinity.

Since cycles are elements of the projective space, the following normalised cycle product:

is more meaningful than the cycle product (7) itself. Note that, [ C₁,C₂ ] is defined only if neither C₁ nor C₂ is a zero-radius cycle (i.e., a point). Also, the normalised cycle product is GL₂(ℂ)-invariant in comparison to the SL₂(ℂ)-invariance in (8).

A reader will instantly recognise the familiar pattern of the cosine of angle between two vectors appeared in (7). Simple calculations show that this geometric interpretation is very explicit in two special cases of interest.

Note the explicit elliptic-hyperbolic analogy between (13) and (14). By the way, both expressions produce real x and y due to inequality (9) for the respective types of pencils. Now we can deduce the following properties of a three-cycle parametrisation.

Proof. The first statement is obvious. For the second we take D_λ(t₀): Λ → Λ which maps C₁∩ C₂ to C₁′∩ C₂′, this transformation maps C_j↦ C_j′ for j=1,2,3. Finally, the last statement follows from (14).

Note that expression (15) is FLT-invariant. Since any loxodrome is an image of SLS under FLT we obtain a three-cycle parametrisation of loxodromes as follows.

Proof. The first statement is obvious, the second follows because properties (1(a)) and (1(b)) are FLT-invariant.

For (3) in the degenerate case C₂=C₃: any M that sends C₂=C₃ to the unit circle will do the job. If C₂≠ C₃ we explicitly describe below the procedure, which produces FLT M mapping the loxodrome to the SLS.

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Figure 15.3: Logarithmic spirals (left) and loxodrome (right) with associated pencils of cycles. This is a combination of Figs. 15.1 and 15.2.

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In essence, the previous proposition says that a three-cycle and (λ,M) parametrisations are equivalent and delivers an explicit procedure producing one from another. However, three-cycle parametrisation is more geometric, since it links a loxodrome to a pair of orthogonal pencils, see Fig. 15.3. Furthermore, cycles C₁, C₂, C₃ (unlike parameters λ and M) can be directly drawn in the plane to represent a loxodrome, which may be even omitted.

15.5 Applications of Three-Cycle Parametrisation

Now we present some examples of the use of three-cycle parametrisation of loxodromes. Any parametrisation mentioned in this paper has some arbitrariness. For pairs (λ′,M) this is described in Proposition 9. Characterisation as orbits from Rem. 10 seems to be most ambiguous: besides the previous freedom in the choice of one-parameter subgroup, we can pick any point of the loxodrome as well. Now we want to resolve non-uniqueness in the three-cycle parametrisation. Recall, that a triple {C₁,C₂,C₃} is non-degenerate if C₂≠ C₃ and C₃ is not zero-radius.

Proof. Necessity of (1) is obvious, since hyperbolic pencils spanned by {C₂,C₃} and {C₂′,C₃′} are both the image of concentric circles centred at origin under the FLT M defining the loxodrome. Necessity of (2) is also obvious since λ′ is uniquely defined by the loxodrome. Necessity of (3) follows from the analysis of the following demonstration of sufficiency.

For sufficiency, let M be FLT constructed through Procedure 15 from {C₁,C₂,C₃}. Then (1) implies that M(C₂′) and M(C₃′) are also circles centred at the origin. Then Lem. 12 implies that the transformation D_{x+i y}(1), where x=[ C₂,C₂′ ] and y=arccos[ C₁,C₁′ ], maps C₁′ and C₂′ to C₁ and C₂, respectively. Furthermore, from identity (14) it follows that the same D_{x+i y}(1) maps C₃′ to C₃. Finally, condition (15) means that x+i (y +2π n)= a(λ′ +2π i) for a=x/λ′ and some n∈ℤ. In other words D_{x+i y}(1)=D_λ′(a), thus D_{x+i y}(1) maps the SLS with parameter λ′ to itself. Since {M(C₁), M(C₂), M(C₃)} and {M(C₁′), M(C₂′), M(C₃′)} are two three-cycle parametrisations of the same SLS, {C₁,C₂,C₃} and {C₁′,C₂′,C₃′} are two three-cycle parametrisations of the same loxodrome.

See Fig. 13.1 for an animated family of equivalent three-cycle parametrisations of the same loxodrome (also posted at [207]). Relation (15), which correlates elliptic and hyperbolic rotations for the loxodrome, regularly appears in this context. The next topic provides another illustration of this.

Proof. Let M be the standard FLT associated with {C₁,C₂,C₃} from Procedure 15. The point C₀ belongs to the loxodrome if the transformation D_λ′(t) for some t maps M(C₀) to the intersection M(C₁)∩ M(C₂). But D_{x+i y}(1) with x= [ C_h,C₂ ] and y=arccos[ C_e,C₁ ] maps M(C_h)→ M(C₂) and M(C_e)→ M(C₁), thus it also maps M(C₀)⊂ M(C_h)∩ M(C_e) to M(C₁)∩ M(C₂). Condition (19) guaranties that D_{x+i y}(1)=D_λ′(x/λ′), as in the previous Prop.

Our final example considers two loxodromes that may have completely different associated pencils.

Proof. A non-degenerate loxodrome intersects any cycle from its hyperbolic pencil with the fixed angle arctan(λ′/(2π)). This is used to amend the intersection angle arccos[ C_h,C_h′ ] of cycles from the respective hyperbolic pencils.

Proof. The first condition simply verifies that C passes C₀, cf. Ex 2. Cycle C, as a degenerate loxodrome, is parametrised by {C_e,C,C}, where C_e is any cycle orthogonal to C and C_e is not relevant in the following. The hyperbolic pencil spanned by two copies of C consists of C only. Thus we put C_h′=C, λ′′=0 in (20) and set that expression equal it to 0 to obtain the second equation in (21).

15.6 Discussion and Open Questions

It was mentioned at the end of Section 15.4 that a three-cycle parametrisation of loxodromes is more geometric than their presentation by a pair (λ,M). Furthermore, three-cycle parametrisation reveals the natural analogy between elliptic and hyperbolic features of loxodromes, see (15) as an illustration. Examples in Section 15.5 show that various geometric questions are explicitly answered in term of three-cycle parametrisation. Thus, our work extends the set of objects in Lie sphere geometry—circles, lines and points—to the natural maximal conformally-invariant family, which also includes loxodromes. In practical terms, three-cycle parametrisation allows one to extend the library figure for Möbuis invariant geometry [211, 212] to operate with loxodromes as well.

It is even more important that the presented technique is another implementation of a general framework [209, 211, 208, 212], which provides a significant advance in Lie sphere geometry. The Poincaré extension of FLT from the real line to the upper half-plane was performed by a pair of orthogonal cycles in [209]. A similar extension of FLT from the complex plane to the upper half-space can be done by a triple of pairwise orthogonal cycles. Thus, triples satisfying FLT-invariant properties (1(a)) and (1(b)) of Prop. 14 present another non-equivalent class of cycle ensembles in the sense of [209]. In general, Lie sphere geometry can be enriched by consideration of cycle ensembles interrelated by a list of FLT-invariant properties [209]. Such ensembles become new objects in the extended Lie spheres geometry and can be represented by points in a cycle ensemble space.

There are several natural directions to extend this work further, here are just few of them:

1. Link our “global” parametrisation of loxodromes with the differential geometric approach from [43, 290, 293]. Our last Cor. 20 can be a first step in this direction.
2. Consider all FLT-invariant non-equivalent classes of three-cycle ensembles on ℂ: pairwise orthogonal triples (representing points in the upper half-space [209]), triples satisfying properties (1(a)) and (1(b)) of Prop. 14 (representing loxodromes), etc.
3. Extend this consideration to quaternions or Clifford algebras [122, 258]. The previous works [291, 292] and availability of FSCc in this setup [65]*Ch. 4 [100] make it rather promising.
4. Consider Möbius transformations in rings of dual and double numbers [198, 191, 209, 190, 203, 196, 261, 19]. There are enough indications that the story will not be quite the same as for complex numbers.
5. Explore further connections of loxodromes with
   • Carleson curves and microlocal properties of singular integral operators [40, 46, 18]; or
   • applications to operator theory [306, 330].

Some combinations of those topics may be fruitful as well.

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1

Of course, this is not the only possible presentation. However, this form is particularly suitable to demonstrate FLT-invariance (8) of the cycle product below.

2

Pedantic consideration of the trivial case ℜ (λ) = ℑ (λ)=0 will be often omitted in the following discussion.

3

Recall that if C₂=C₃, then SLS is degenerate and coincides with C₂=C₃.